Bivariate extensions of the Extended Quadrature Method of Moments (EQMOM) to describe coupled droplet evaporation and heat-up

In this study, four Eulerian moment methods based on the Extended Quadrature Method of Moments (EQMOM) ( Yuan, Laurent, & Fox, 2012 ) are applied to model the evaporation of droplets. EQMOM enables the number density function (NDF) to be reconstructed from a set of moments of a droplet ensemble. Knowledge of the NDF allows us to describe the disappearance fl ux of the smallest droplets accurately, which is a critical issue in moment methods ( Fox, Laurent, & Massot, 2008 ; Yuan et al., 2012 ). As EQMOM is restricted to univariate distributions, initially only the droplet size is considered as an internal coor- dinate. The description is then extended to the bivariate models mono-thermal Extended Conditional Quadrature Method of Moments (mt-ECQMOM), whose formulation is based on the semi-kinetic approach ( Vie, Laurent, & Massot, 2013 ) and Extended Conditional Quadrature Method of Moments (ECQMOM, Marchisio & Fox, 2013 ). Based on ECQMOM and the general principle of a sectional method, a third bivariate approach is proposed, called the Extended Conditional Sectional Method of Moments (ECSQMOM). The resulting three bivariate approaches allow us to describe the coupling of heat-up and evaporation using the internal coordinates droplet diameter and temperature. The results based on EQMOM, mt-ECQMOM, ECQMOM and ECSQMOM are evaluated against a direct numerical solution of the population balance equation (PBE) investigating a test setup based on experimental studies for iso-octane sprays. Results indicate that in the limiting case of a mono-thermal distribution, the ECQMOM model implementation referenced above is not applicable to describe evaporating sprays accurately, whereas the mt-ECQMOM leads to much better results. ECSQMOM is also found to yield good results, even if the accuracy of mt-ECQMOM is not reached. However, since ECSQMOM offers the potential to capture distributions with nonzero conditional variance, it can be considered a suitable approach to describe the evolution of droplet distribution in sprays in future applications.

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